Chapter 11 · Volume
Volume of Boxes: From Packing to Multiplying
Pack a box with unit cubes one layer at a time, then see why the count matches multiplying its edge lengths in any grouping.
In this lesson you find the volume of a box by packing it with unit cubes. Then you see why that count is a multiplication.
Recall that volume is the number of unit cubes that fill a solid with no gaps or overlaps. A centimetre cube is a unit cube that is 1 cm long on every edge. Its volume is 1 cubic centimetre.
The boxes in this lesson are right rectangular prisms. These are solids with six rectangle faces that meet at square corners, like a shoebox. A box's size is given by three edge lengths. When box A sits upright in front of you, its length runs from left to right, its width runs from front to back, and its height runs from bottom to top. Box A is 4 cm long, 3 cm wide and 2 cm high.
Packing a box means filling it completely with unit cubes, with no gaps and no overlaps. You can pack a box one layer at a time. A layer is a flat slab of cubes, one cube thick. Start with front-to-back layers. Each one faces you, and they stand one behind another from the front of the box to the back.
The figure below shows one front-to-back layer of box A on its own, as a box just one cube thick. Each square on its front is the face of one centimetre cube. Each cube is 1 cm on a side. So the 4 cubes across match the 4 cm length, and the 2 cubes up match the 2 cm height. This front-to-back layer holds cubes.
Each front-to-back layer is 1 cm thick, and box A is 3 cm from front to back. So box A holds 3 front-to-back layers, one behind another. Each one is the same size as the front-to-back layer you just counted, so each one holds 8 cubes.
The next figure shows the whole box. It draws the front-to-back layer that faces you cube by cube. The cubes behind it are hidden. They are still there, because the box is completely packed. Look at the top or the right side. Each strip there is one whole front-to-back layer, and there are 3 strips.
Now count one front-to-back layer at a time: 8, 16, 24. Box A holds 24 centimetre cubes, so its volume is 24 cubic centimetres.
Adding the same number again and again is multiplying. Three front-to-back layers of 8 cubes is . So you can write the count as the number of cubes in one layer times the number of layers. All the layers must run in the same direction.
Now see where those numbers come from. The 8 is : 4 cubes for the 4 cm length times 2 cubes for the 2 cm height. The 3 front-to-back layers come from the 3 cm width. So the count is , the product of the three edge lengths of box A. Packing and multiplying give the same volume.
This works for any right rectangular prism whose edge lengths are whole numbers of one unit, packed with unit cubes of that unit. Each cube is 1 unit on a side, so the number of cubes along each edge equals that edge's length.
Worked example
Counting a box of toy blocks
A box is packed full of toy blocks, with no gaps or overlaps. Each block is a 1-inch cube. The box is 5 inches long, 2 inches wide and 3 inches high. Count the blocks layer by layer to find the volume. Then check by multiplying the edge lengths.
Count one front-to-back layer. Each block is 1 inch on a side. So the 5-inch length holds 5 blocks across, and the 3-inch height holds 3 blocks up.
The box is 2 inches from front to back, so it holds 2 front-to-back layers of 15 blocks. Adding equal front-to-back layers is the same as multiplying the cubes in one layer by the number of layers.
Check with the edge lengths. The 15 came from the length times the height, and the 2 is the width. So the count is the product of all three edge lengths.
Answer
The box holds blocks, so its volume is cubic inches. Packing and multiplying the edge lengths give the same volume.
A box can be cut into layers in any of its three directions. Each direction counts the same cubes, so each gives the same volume. Bottom layers lie flat like the bottom of the box and stack from the bottom up.
The next figure shows the same box A, turned so its bottom faces you. Now the 4 cm length runs across, the 3 cm width runs up, and the 2 cm height runs from front to back. Turned this way, the bottom layers stand one behind another. The one drawn cube by cube at the front holds length times width, cubes. The box is 2 cm high, so it has 2 bottom layers, shown by the 2 strips on top. Its volume is cubic centimetres.
Compare this with the front-to-back count, . The factors, the numbers you multiply, are the same. The order changed: 3 now comes before 2. The grouping is which two factors you multiply first, and it changed too: 4 now pairs with 3, not with 2. The volume is still 24 cubic centimetres, because it is the same box with the same cubes.
The third direction gives side layers. These stand upright like the right side of the box and line up from left to right. In box A, each side layer is 3 cubes from front to back (the 3 cm width) and 2 cubes high (the 2 cm height): cubes. Each side layer is 1 cm thick, and the box is 4 cm long, so it has 4 side layers. Writing the 4 side layers first, the volume is cubic centimetres.
Put the bottom-layer count beside it: . The order 4, 3, 2 stays the same, and only the grouping changes. Grouping 4 and 3 first counts bottom layers. Grouping 3 and 2 first counts side layers. Both ways count every cube of the same box exactly once, so both give 24.
So for three whole numbers that are not zero, grouping them differently cannot change the product. Read them as the edge lengths of a box, and every grouping counts the same cubes of that box.
Key concept
Packing is multiplying
Take a right rectangular prism with whole-number edge lengths. Pack it with unit cubes of the same unit as its edges, with no gaps or overlaps. Cut it into layers that all run in one direction: front-to-back, bottom or side. Each of those layers holds the same number of cubes, so the volume is the number of cubes in one layer times the number of layers. Both numbers come from the edge lengths, so the volume is also the product of the three edge lengths, in cubic units. Whatever direction, order or grouping you choose, you count the same cubes in the same box, so you get the same volume.
Worked example
Reading a product of three numbers as a volume
Read as the volume of a box. Find the volume. Then find it again as , and explain why the two groupings agree.
Read the factors as edge lengths. Picture a box 7 units long, 5 units wide and 2 units high, completely packed with unit cubes. The parentheses group the 7 and the 5, so first find length times width, the number of cubes in one bottom layer.
The box is 2 units high, so it has 2 bottom layers of 35 cubes each.
Now group the 5 and the 2 first. Width times height is the number of cubes in one side layer, a friendly 10. The box is 7 units long, so it has 7 side layers. Write the 7 first to keep the order 7, 5, 2.
Compare. The order 7, 5, 2 did not change, and only the grouping did. Both ways count every cube of the same box exactly once, so they must give the same number.
Answer
The box has a volume of cubic units. The groupings and both give because they count the same cubes in the same box.
Takeaway: To find the volume of a box completely packed with unit cubes, count the cubes in one layer. Then multiply by the number of layers in that same direction. When the edge lengths are whole numbers of the cube's unit, that is the same as multiplying the three edge lengths. You can change the order or the grouping of those three factors, because every way counts the same cubes in the same box.
Next, you will write this idea as formulas for volume. One formula uses the length, width and height. The other uses the area of the base times the height.